Chapter 5: Trigonometric Functions

5-1: Trigonometric Ratios in Right Triangles

Trigonometry - study of the circular functions and the relationships between sides and angles of a right triangle
Trigonometry Functions - ratios of the sides of a right triangle
Angle of Elevation - starts at horizon and goes in positive direction
Angle of Depression - starts at horizon and goes in negative direction
This first lesson of chapter 5 deals with the trig ratios in right triangles. Sine of a theta of a right triangle=the opposite side divided by the hypotenuse, cosine of a theta=adjacent side divided by the hypotenuse, and tangent of a theta=the opposite side divided by the adjacent side. We can find out angles and sides of a triangle by using the trigonometric ratios, as long as certain sides or angle measures are given. Also, a degree can be divided into measures called minutes and seconds. One degree=60 minutes, and one minute=60 seconds.


-Nathan F.

As though we've gone through the majority of the year without knowing the true meaning of trigonometry, we learned that the subject means "the study of the circular functions and the relationships between sides and angles of a right triangle."
Also, some new functions are secant which is the inverse of cosine, cosecant which is the inverse of sine, and cotan which is the inverse of tangent. Although they are not exactly useful at this moment, they are sure to be applied in the future.

this is an example of elevation, it starts at the horizon and goes in a positive direction upwards.

this is an example of depression, it starts at the horizon and goes in a negative direction.
minute-
A unit of angle measure equal to of a degree. There are 60 minutes in one degree. Minutes are indicated using
the ' symbol, so 12°45' means 12 degrees and 45 minutes, or 12.75 degrees.
second-
A unit of angle measure equal to of a minute. There are 60 seconds in one minute and 3600 seconds in one degree. Seconds are indicated using the " symbol, so 12°45'33" means 12 degrees, 45 minutes, and 33 seconds, or degrees.

The pictures and definitions for minute and second come from the website http://www.mathwords.com/s/second.htm.
Andrea M.

5-2: The Law of Cosines

When you cannot use the Law of Sine to find the angles or sides of a triangle, the Law of Cosines is another option. This equation is based off of the Pythagorean Theorem (a²+b²=c²)
Definition: For any triangle ABC,
c²=a²+b²-2abcosC
*this does not need to be a right triangle for this to work.
By plugging in the information you do have about the triangle into this equation you can then find what you are missing. The reason you do not need a 90° angle in your triangle for this equation is because when you use this equation with a right triangle the Cosine of 90° ends up being 0, which cancels the end part of that equation out, leaving just the pyhagorean theorem.

~Shelby N.


Jeremy G.
In this lesson we are introduced to the law of cosines. This theorem states that in any triangle ABC, c^2 = a^2 + b^2 - 2abcos(C). The variable c stands for the length of the side you are looking for. a and b are the lengths of the other two sides. C stands for the included angle. Depending on the problem either c or C could be given. This theorem is useful when you are given two sides and the included angle of a triangle to help you find the length of the third side. This is known as the SAS condition. It can also be used to find the angles of a triangle when you only know the length of the sides (SSS condition)

5-3: The Inverse Cosine Function

The inverse must be a function by passing the horizontal line test. If it doesn't you need to put a restriction on the domain of y=cos x. We also need to get all possible output values. The range of the outputs should be -1<=y<=1.
The Inverse Cosine Function-y=cos^-1 x,FF.
Toots(Dalton)

Three criteria for an appropriate domain:
1. The domain should include the angles between 0 and pi - this is because they are the measure of the acute angles of a right triangle
2. On the restricted domain, the function should take on all the values of the range, which is all real numbers from -1 to 1
3. If possible, the function should be continuous in the restricted domain.

Arccos is sometimes used instead of cos^-1
Mary
Some info on 5-3

5-4: The Law of Sines

Page 327-330
When you cannot find the sides or angles in a triangle by using the Laws of Cosines, you can try using the Law of Sines. A simple theorem states the Law of Sines as in any triangle ABC, sin A/a= sin B/b= sin c/c. By using this theorem, you can find the length of a second side of a triangle given the measures of two angles and a side (ASA or AAS conditions). The Law of Sines can also be used in order to determine the measure of a second angle of a triangle when two sides and a nonincluded angle are known (SSA condition). Another use for the law is when the side opposite the given angle is larger than the other given side (SsA condition). In conclusion, the Law of Sines is useful for ASA, AAS, SSA, and SsA conditions.
Lesson 5-4 also mentions another theorem: SAS Area Formula for a Triangle. This theorem states that in any triangle the area is one-half the product of the lenghts of any two sides and the sine of their included angle. For any triangle ABC, the area K can be given by two formulas: K=1/2absinC or K=1/2bcsinA.
-------------------------------------------------------------------------------------------------Zach G.-----------------------------------------------------------------------------------------------------------------
SINE
Law of Sines: Triangle ABC- SIN A/a=SIN B/b=SIN C/c (use to find the length of a side of a triangle, in order to use this theorem you need to know either an angle measure and the length of two sides or the measure of two angles and th length of one side)

SAS Area Formula: Triangle ABC- area(K) K=1/2absinC or 1/2bcsinA

Nate M.

More from Mr. Lamb:
Law of Sines
Interactive Practice (Law of Sines, Law of Cosines, triangles)

5-5: The Inverse Sine Function

Inverse Sine Function- y=sin^-1 x
toots(Dalton)

Arcsin is sometimes used instead of sin^-1
The domain of the Inverse Sine Function
It is possible to restrict the domain of a sine function so that the inverse will then be a function

Mary
Some info on 5-5

5-6: The Inverse Tangent Function

Inverse Tangent Function- y=tan^-1 x, FF.
Toots(Dalton)

Possible to restrict domain to make the inverse a function
Arctan is sometimes used in place of tan^-1

Mary
You can use a calculator to estimate all values of the inverse tan funtion
Some info on 5-6

5-7: General Solutions to Trigonometric Equations

A Trigonometric Equation is an equation where the variable is within one of the trig functions. The three types of domains that Trigonometric equations can have are:

1. Restricted domains
2. One period
3. All real numbers

A General Solution is a solution that takes into account all possible solutions for a trig equation. To find a general solution to an equation you add or subtract the period to or from the solution(s) to the equation. For example:
If the solutions you found for an equation with a period of 2π were x = .373 and x = 5.678 the general solution to the equation would be x = .373 + 2πn and x = 5.678 + 2πn where n is an integer representing the number of periods.
-Dan H.

A trigonomic function is function where the independent variable is in the parentheses of a cosine, sine, tangent, or any other trig. function. Generally, there are three different domains: restricted domains that are used to graph the inverse of the three main trig. functions; the domain equal to one period; and all real numbers.

In this lesson, we focused on the 3rd domain, that of all real numbers, and considered an equation for a general solution, or an equation which gives all solutions to trigonomic equations.

To find a general equation, find first the smallest positive answer or largest negative answer to the trigonomic function. For instance, sin (theta) = 2, solve sin-1(2) to get (theta) by itself. This will give you one answer. To set it up from here, add sin-1(2) to the period of the sine function, 2(pi) times n, or the set of integers. Therefore, (theta) = sin-1(2) + 2(pi)n. But there is another set of solutions too. (pi) - sin-1(2) will give you a different number, but a number that will work nonetheless. So, (theta) also equals ((pi)-sin-1(2)) + 2(pi)n.

Secondly, we learned about quadratic equations with trig. functions. Substitute a variable in for let's say tan of x (or sin x or cos x depending on which trig. function you use). Then factor and set each set of parentheses to zero. Whatever answer you get, set to tan x (or sin x or cos x depending on which trig. function you use). This should give you the answer.

-Robert M.

5-8: From Washington to Beijing

ERROR ALERT: Slide show is updated to reflect the fixes for Example 3. I used the incorrect formula in the example, using sinC instead of cosC at the end! The correct formula is now in the slides.


In this lesson we learned how to find distances between points on the same longitude and latitude. We also learned how to find distances between any two points on a sphere.
To find the distance between two points on the same longitude, you want to find the minor arc of the great circle connecting the two points. When you find the minor arc, you divide the degrees of the angle over 360 degrees. Then you multiply it by 2, pie, and 3960 to get your final answer in miles.
When finding the distance between two different points on a sphere, you have to use the Spherical Law of Cosines.
If ABC is a spherical triangle with sides a,b, and c, then
cosc=cos acos b + sin a sin b cos C
Which means you have to form a triangle and find the angles you need to fill into the Spherical law of Cosines. Once you figure that out, take your answer and multiply it by 2, pie, and 3960. Your final answer will be in miles.

Kelly H.

The fist thing we learned in this lesson was that the shortest distance from one point to another on a sphere lies along a great circle. A Great Circle of a sphere is the intersection of a sphere and a plane containing the center of the sphere. The great circle then has the same center as the sphere, thus having the same radius and circumference as the sphere. We also learned that on Earth, a meridian is also known as a line of longitude and is a semicircle whose endpoints are the north and South Pole. The meridian on Earth which is 180 deg. E and 180 deg. W is called the International Date Line.

There are 3 different possibilities for two points to be on the Earth in relation to each other, and we found out how to find the distance between the points in all 3 different situations:
- Points on the same Longitude
- Points on the same Latitude
- Any two points on a sphere that are not on the same
longitude or latitude

It is quite simple to find the shortest distance between two points on the same latitude. All that must be done to find the distance is to subtract the latitudes of the two points and then divide it by 360 degrees. Multiply that answer by 2*pi*3960(radius of the earth) and you should get the arc length of the sector which is formed by the center of the earth and the two points. The arc length you found is the distance between the two points on the same longitude.

You can not use the same method as above in order to find the shortest distance between two points on the same latitude. This is because all meridians are great circles, but only one line of latitude is a great circle (the equator), and the answer to the shortest distance between two points on a sphere is always a great circle. In order to find the shortest distance between two points on the same latitude, and any two points on a sphere, you have to use the same formula (The SPHERICAL LAW OF COSINES). The Spherical Law of Cosines is as follows: If ABC is a spherical triangle with sides a,b, and c, then cosc=cosa+cosb+sina*sinb*cosC. The spherical law of cosines finds great circle distances, so this is why you must use the spherical law of cosines to find the shortest distance between two points not on the same longitude.

-Austin R.

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